Probability and statistics
What you must understand
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). For equally likely outcomes, the probability of an event is the number of favourable outcomes divided by the total number of outcomes. You must be comfortable with the complement rule (P(not A) = 1 − P(A)), with probabilities expressed as fractions, decimals or percentages, and with the idea that the probabilities of all mutually exclusive outcomes sum to 1.
Know how to list a sample space systematically - a table, list or tree diagram - for two-step experiments such as tossing two coins or spinning two spinners. Understand independent events, where the outcome of one does not change the probability of the other, and be able to find the probability of two independent events both happening by multiplying. Recognise when relative frequency from data (experimental probability) is being used to estimate a theoretical probability, and understand that larger samples give more reliable estimates.
Traps to watch
- Writing a probability greater than 1 or a negative probability - always check the answer lies between 0 and 1.
- Mixing up "at least one" with "exactly one"; "at least one" is usually easiest through the complement.
- Forgetting that the sample space must list all equally likely outcomes, and miscounting it (two coins give four outcomes, not three).
- Treating events as independent when the first outcome removes an option (drawing without replacement).
- Assuming a small experimental sample must match the theoretical probability exactly.
Worked example
Two fair coins are tossed. The sample space is HH, HT, TH, TT, four equally likely outcomes. P(exactly one head) = 2/4 = 0.5. P(at least one head) = 1 − P(no heads) = 1 − 1/4 = 3/4. Rule of thumb: for "at least one", find the probability of none and subtract from 1.